256 research outputs found

    Polymeric Phase of Simplicial Quantum Gravity

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    We deduce the appearance of a polymeric phase in 4-dimensional simplicial quantum gravity by varying the values of the coupling constants and discuss the geometric structure of the phase in terms of ergodic moves. A similar result is true in 3-dimensions.Comment: 6 pages, revte

    Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

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    We consider a lattice gas on the discrete d-dimensional torus (Z/NZ)d(\mathbb{Z}/N\mathbb{Z})^d with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field E/N. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., E constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field E is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on E.Comment: Published in at http://dx.doi.org/10.1214/11-AAP805 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Large deviations of the empirical flow for continuous time Markov chains

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    We consider a continuous time Markov chain on a countable state space and prove a joint large deviation principle for the empirical measure and the empirical flow, which accounts for the total number of jumps between pairs of states. We give a direct proof using tilting and an indirect one by contraction from the empirical process.Comment: Minor revision, to appear on Annales de l'Institut Henri Poincare (B) Probability and Statistic

    Dynamical Phase Transitions for Fluxes of Mass on Finite Graphs

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    We study the time-averaged flux in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flux is given by a variational formulation involving paths of the density and flux. We give sufficient conditions under which the large deviations of a given time averaged flux is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph

    Large deviations for a stochastic model of heat flow

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    We investigate a one dimensional chain of 2N2N harmonic oscillators in which neighboring sites have their energies redistributed randomly. The sites −N-N and NN are in contact with thermal reservoirs at different temperature τ−\tau_- and τ+\tau_+. Kipnis, Marchioro, and Presutti \cite{KMP} proved that this model satisfies {}Fourier's law and that in the hydrodynamical scaling limit, when N→∞N \to \infty, the stationary state has a linear energy density profile θˉ(u)\bar \theta(u), u∈[−1,1]u \in [-1,1]. We derive the large deviation function S(θ(u))S(\theta(u)) for the probability of finding, in the stationary state, a profile θ(u)\theta(u) different from θˉ(u)\bar \theta(u). The function S(θ)S(\theta) has striking similarities to, but also large differences from, the corresponding one of the symmetric exclusion process. Like the latter it is nonlocal and satisfies a variational equation. Unlike the latter it is not convex and the Gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state. We also briefly discuss more general model and find the features common in these two and other models whose S(θ)S(\theta) is known.Comment: 28 pages, 0 figure

    Box-ball system: soliton and tree decomposition of excursions

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    We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514–3519, 1990). Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari et al. (Soliton decomposition of the box-ball system (2018). arXiv:1806.02798) proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020) propose a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In the present paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall (Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85, vol. 1204, pp. 447–464. Lecture Notes in Mathematics. Springer, Berlin (1986)). A ball configuration distributed as independent Bernoulli variables of parameter λ < 1∕2 is in correspondence with a simple random walk with negative drift 2λ − 1 and having infinitely many excursions over the local minima. In this case the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020). We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Gabrielli, Davide. Universita degli Studi dell'Aquila; ItaliaXIII Symposium on Probability and Stochastic ProcessesMéxicoUniversidad Nacional Autónoma de Méxic
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